gausslemma2d - Metamath Proof Explorer

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Theorem gausslemma2d 27128

Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
gausslemma2d.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))
gausslemma2d.n	⊢ 𝑁 = (𝐻 − 𝑀)

**Assertion**
Ref	Expression
gausslemma2d	⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑥,𝑀 Allowed substitution hints: 𝑅(𝑥) 𝑁(𝑥)

**Proof of Theorem gausslemma2d**
Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3		gausslemma2d.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
4		gausslemma2d.m	. . 3 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
5		gausslemma2d.n	. . 3 ⊢ 𝑁 = (𝐻 − 𝑀)
6	1, 2, 3, 4, 5	gausslemma2dlem7 27127	. 2 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1)
7		eldifi 4126	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16618	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	8	nnred 12234	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
10		prmgt1 16641	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
11	9, 10	jca 511	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
12	1, 7, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
13		1mod 13875	. . . . . 6 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (1 mod 𝑃) = 1)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → (1 mod 𝑃) = 1)
15	14	eqcomd 2737	. . . 4 ⊢ (𝜑 → 1 = (1 mod 𝑃))
16	15	eqeq2d 2742	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 ↔ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)))
17		neg1z 12605	. . . . . . . . . . 11 ⊢ -1 ∈ ℤ
18	1, 4, 2, 5	gausslemma2dlem0h 27117	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		zexpcl 14049	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (-1↑𝑁) ∈ ℤ)
20	17, 18, 19	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑𝑁) ∈ ℤ)
21		2nn 12292	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
23	1, 2	gausslemma2dlem0b 27111	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ∈ ℕ)
24	23	nnnn0d 12539	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
25	22, 24	nnexpcld 14215	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝐻) ∈ ℕ)
26	25	nnzd 12592	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐻) ∈ ℤ)
27	20, 26	zmulcld 12679	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℤ)
28	27	zred 12673	. . . . . . . 8 ⊢ (𝜑 → ((-1↑𝑁) · (2↑𝐻)) ∈ ℝ)
29		1red 11222	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
30	28, 29	jca 511	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ))
32	1	gausslemma2dlem0a 27110	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
33	32	nnrpd 13021	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
34	20, 33	jca 511	. . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
35	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺))
36		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃))
37		modmul1 13896	. . . . . 6 ⊢ (((((-1↑𝑁) · (2↑𝐻)) ∈ ℝ ∧ 1 ∈ ℝ) ∧ ((-1↑𝑁) ∈ ℤ ∧ 𝑃 ∈ ℝ⁺) ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
38	31, 35, 36, 37	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ (((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃)) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃))
39	38	ex 412	. . . 4 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃)))
40	20	zcnd 12674	. . . . . . . . 9 ⊢ (𝜑 → (-1↑𝑁) ∈ ℂ)
41	25	nncnd 12235	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐻) ∈ ℂ)
42	40, 41, 40	mul32d 11431	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)))
43	18	nn0cnd 12541	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℂ)
44	43	2timesd 12462	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
45	44	eqcomd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 𝑁) = (2 · 𝑁))
46	45	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = (-1↑(2 · 𝑁)))
47		neg1cn 12333	. . . . . . . . . . . 12 ⊢ -1 ∈ ℂ
48	47	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → -1 ∈ ℂ)
49	48, 18, 18	expaddd 14120	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
50	18	nn0zd 12591	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
51		m1expeven 14082	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(2 · 𝑁)) = 1)
53	46, 49, 52	3eqtr3d 2779	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑𝑁) · (-1↑𝑁)) = 1)
54	53	oveq1d 7427	. . . . . . . 8 ⊢ (𝜑 → (((-1↑𝑁) · (-1↑𝑁)) · (2↑𝐻)) = (1 · (2↑𝐻)))
55	41	mullidd 11239	. . . . . . . 8 ⊢ (𝜑 → (1 · (2↑𝐻)) = (2↑𝐻))
56	42, 54, 55	3eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) = (2↑𝐻))
57	56	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((2↑𝐻) mod 𝑃))
58	40	mullidd 11239	. . . . . . 7 ⊢ (𝜑 → (1 · (-1↑𝑁)) = (-1↑𝑁))
59	58	oveq1d 7427	. . . . . 6 ⊢ (𝜑 → ((1 · (-1↑𝑁)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
60	57, 59	eqeq12d 2747	. . . . 5 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) ↔ ((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
61	2	oveq2i 7423	. . . . . . . 8 ⊢ (2↑𝐻) = (2↑((𝑃 − 1) / 2))
62	61	oveq1i 7422	. . . . . . 7 ⊢ ((2↑𝐻) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃)
63	62	eqeq1i 2736	. . . . . 6 ⊢ (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃))
64		2z 12601	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
65		lgsvalmod 27070	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
66	64, 1, 65	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → ((2 /_L 𝑃) mod 𝑃) = ((2↑((𝑃 − 1) / 2)) mod 𝑃))
67	66	eqcomd 2737	. . . . . . . 8 ⊢ (𝜑 → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((2 /_L 𝑃) mod 𝑃))
68	67	eqeq1d 2733	. . . . . . 7 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) ↔ ((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃)))
69	1, 4, 2, 5	gausslemma2dlem0i 27118	. . . . . . 7 ⊢ (𝜑 → (((2 /_L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
70	68, 69	sylbid 239	. . . . . 6 ⊢ (𝜑 → (((2↑((𝑃 − 1) / 2)) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
71	63, 70	biimtrid 241	. . . . 5 ⊢ (𝜑 → (((2↑𝐻) mod 𝑃) = ((-1↑𝑁) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
72	60, 71	sylbid 239	. . . 4 ⊢ (𝜑 → (((((-1↑𝑁) · (2↑𝐻)) · (-1↑𝑁)) mod 𝑃) = ((1 · (-1↑𝑁)) mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
73	39, 72	syld 47	. . 3 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = (1 mod 𝑃) → (2 /_L 𝑃) = (-1↑𝑁)))
74	16, 73	sylbid 239	. 2 ⊢ (𝜑 → ((((-1↑𝑁) · (2↑𝐻)) mod 𝑃) = 1 → (2 /_L 𝑃) = (-1↑𝑁)))
75	6, 74	mpd 15	1 ⊢ (𝜑 → (2 /_L 𝑃) = (-1↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ifcif 4528 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 1c1 11117 + caddc 11119 · cmul 11121 < clt 11255 − cmin 11451 -cneg 11452 / cdiv 11878 ℕcn 12219 2c2 12274 4c4 12276 ℕ₀cn0 12479 ℤcz 12565 ℝ⁺crp 12981 ...cfz 13491 ⌊cfl 13762 mod cmo 13841 ↑cexp 14034 ℙcprime 16615 /_L clgs 27048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-ioo 13335 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-prod 15857 df-dvds 16205 df-gcd 16443 df-prm 16616 df-phi 16706 df-pc 16777 df-lgs 27049

This theorem is referenced by: 2lgs 27161

W3C validator